Bake per vertex color to Texture
The issue is that .OBJ format by standard does not support vertex colors, so Blender isn't importing them. The way to append vertex color after vertex position is only a third-party hack to the format, and thus the support is limited in many applications.

Please follow this post about importing an .OBJ file with vertex colors into Blender and you should be able to see, render and bake the colors no problem.

How to display and use Vertex Color from OBJ model